Problem 30 - Physics

Question

The inner and outer surfaces of a cell membrane carry a negative and positive charge, respectively. Because of these charges, a potential difference of about \(0.070 \mathrm{~V}\) exists across the membrane. The thickness of the cell membrane is \(8.0 \times 10^{-9} \mathrm{~m}\). What is the magnitude of the electric field in the membrane?

Step-by-Step Solution

Answer

The electric field is approximately \(8.75 \times 10^6 \mathrm{~V/m}\).

Step 1: Understanding the Problem

We need to find the magnitude of the electric field between two charges on a cell membrane. The voltage (potential difference) across the membrane is given as \(0.070 \mathrm{~V}\), and the thickness of the membrane is \(8.0 \times 10^{-9} \mathrm{~m}\).

Step 2: Using the Formula for Electric Field

The electric field \(E\) between two points with potential difference \(V\) and separation \(d\) is given by the formula: \[ E = \frac{V}{d} \]

Step 3: Plugging in the Values

Substitute \(V = 0.070 \mathrm{~V}\) and \(d = 8.0 \times 10^{-9} \mathrm{~m}\) into the formula to get: \[ E = \frac{0.070 \mathrm{~V}}{8.0 \times 10^{-9} \mathrm{~m}} \]

Step 4: Calculating the Electric Field

Perform the division: \[ E = \frac{0.070}{8.0 \times 10^{-9}} \approx 8.75 \times 10^6 \mathrm{~V/m} \]

Step 5: Conclusion

The magnitude of the electric field in the membrane is \(8.75 \times 10^6 \mathrm{~V/m}\).

Key Concepts

Cell Membrane

The cell membrane is an essential part of all living cells, acting as a barrier that separates and protects the internal environment of the cell from the external surroundings. It is a dynamic structure composed predominantly of a double layer of lipids, embedded with proteins.
These proteins play various roles, including transporting substances in and out of cells and communicating with other cells. The cell membrane is crucial for maintaining the integrity of the cell and enabling specific processes necessary for life.
The electrical properties of the cell membrane are vital as they allow for the generation of electrical signals used in cell communication, particularly in nerve cells. This is achieved by the differential distribution of electrical charges across the inner and outer surfaces. When studying topics like electric fields related to the cell membrane, we look into these charge distributions and their influences.

Potential Difference

A potential difference, commonly referred to as voltage, is the difference in electric potential between two points. In the context of cell membranes, it is the electrical charge difference between the interior and exterior surfaces of the membrane.
This difference in charge creates what we call "membrane potential." Typically, the inside of the cell is negatively charged when compared to the outside. Membrane potentials are crucial in the function of neurons and muscle cells, as they enable the transmission of signals.
The potential difference across a cell membrane is much smaller than what we encounter with household electrical systems, typically around 0.070 V. This small voltage, however, is significant when observed at the microscopic scale of the cell membrane, impacting the behavior of ions and electrical signals.

Voltage

Voltage is a measure of electrical potential energy per unit charge. In cell biology, voltage is synonymous with the membrane potential, the numerical measure of electrical potential difference across the cell membrane.
Voltage across cell membranes affects how the cell interacts with its environment and responds to different stimuli. It can drive the transport of ions across the membrane, serve as a signal in nerve transmission, and initiate cellular processes.
In practical terms, when calculating electric fields related to the cell membrane, voltage is a key component of the formula used. It helps us understand how charges within and outside the cells interact, which is crucial for biological functions.

Membrane Thickness

Membrane thickness refers to the physical distance between the inner and outer surfaces of the cell membrane. For most cells, this thickness is typically around 8.0 × 10^{-9} meters, an extremely thin barrier.
But even this thin layer plays a significant role in forming potential differences and electric fields. The thickness of the membrane affects the magnitude of the electric field present, as observed in the formula for the electric field: \[ E = \frac{V}{d} \]where \(E\) is the electric field, \(V\) is the potential difference, and \(d\) is the membrane thickness.
This thickness must be precisely maintained for the cell to function correctly. Variations can influence the cell's ability to operate and interact properly due to the resulting changes in membrane potential and electric field strength within the membrane.